Disconnectedness of the subgraph "Equation missing" for the group "Equation missing"

Abstract

In this paper we show that the subgraph F 3 is disconnected and that for all integers m, we find all integers a and b such that (9m 2 -4)a 2 + 4 and 5b 2 ± 4 are square. It turns out that the set of numbers b comprises the Fibonacci numbers.

Solutions to some congruence equations via suborbital graphs

“… 1991 ), some other finitely generated groups have been studied by suborbital graphs (see Akbaş and Başkan 1996 ; Akbaş 2001 ; Akbaş et al. 2013 ; Beşenk et al. 2013 ; Deger et al.…”

“… 2015 ); The set of vertices of some suborbital graphs is strongly connected to the Fibonacci sequence (Akbaş et al. 2013 ). In this light, we conclude that these graphs might be worth examining when just viewed from number theory aspect.…”

“…The set of vertices of some suborbital graphs is strongly connected to the Fibonacci sequence (Akbaş et al. 2013 ).…”

Suborbital graphs for the Atkin--Lehner group

“…Elliptic elements do not necessarily correspond to circuits of the same order. On the other hand, it is worth noting that these graphs give some number theoretical results about continued fractions and Fibonacci numbers as in [4,8,17].…”

“…Then similar studies were done for related finitely generated groups.The reader is referred to [2][3][4][5]8,9,[11][12][13][14][15][16] for some relevant previous work on suborbital graphs. Firstly, in [3], it was proved that the elliptic elements in Γ 0 (n) correspond to circuits in the subgraph F u,n of the same order and vice versa.…”